1.3 Matrices and Matrix Operations 39 Working with Proofs 35. Prove: If A and B are n x n matrices, then tr(A + B) = tr(A) + tr(B) 36. a. Prove: If AB and BA are both defined, then AB and BA are square matrices. b. Prove: If A is an m x n matrix and A(BA) is defined, then B is an n x m matrix.
1.3 Matrices and Matrix Operations 39 Working with Proofs 35. Prove: If A and B are n x n matrices, then tr(A + B) = tr(A) + tr(B) 36. a. Prove: If AB and BA are both defined, then AB and BA are square matrices. b. Prove: If A is an m x n matrix and A(BA) is defined, then B is an n x m matrix.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Questions 35, 36 on paper please
![### Working with Proofs
**35.** Prove: If \( A \) and \( B \) are \( n \times n \) matrices, then
\[ \text{tr}(A + B) = \text{tr}(A) + \text{tr}(B) \]
**36. a.** Prove: If \( AB \) and \( BA \) are both defined, then \( AB \) and \( BA \) are square matrices.
**b.** Prove: If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times m \) matrix and \( A(BA) \) is defined, then \( B \) is an \( n \times m \) matrix.
### True-False Exercises
**TF.** In parts (a) - (o) determine whether the statement is true or false.
---
#### Explanation of Notation and Concepts:
- **Matrices:** Rectangular arrays of numbers arranged in rows and columns.
- **Square Matrix:** A matrix with the same number of rows and columns, i.e., an \( n \times n \) matrix.
- **Trace (tr):** The sum of the elements on the main diagonal of a square matrix.
##### Detailed Explanation of Problems:
- **Problem 35:** This involves proving that the trace of the sum of two \( n \times n \) matrices is equal to the sum of their traces. This is a fundamental property of the trace function in linear algebra.
- **Problem 36a:** This problem requires proving that if the products \( AB \) and \( BA \) are defined (meaning matrix multiplication is possible), both resulting matrices will be square matrices.
- **Problem 36b:** This involves proving a necessary condition for the dimensions of matrix \( B \) when matrix \( A \) is \( m \times n \), \( B \) is an \( n \times m \) matrix, and the product \( A(BA) \) is defined. This explores the implications of matrix multiplication on the dimensions of the matrices involved.
These exercises aim to deepen the understanding of the properties of matrix operations and the implications of these operations on the dimensions and structure of matrices.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F48922dd6-b6f8-4fac-84c6-05a7db5750f6%2Ff8436c2a-6f8f-44d6-a1b3-0084d8674eb9%2Ftjvic22_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Working with Proofs
**35.** Prove: If \( A \) and \( B \) are \( n \times n \) matrices, then
\[ \text{tr}(A + B) = \text{tr}(A) + \text{tr}(B) \]
**36. a.** Prove: If \( AB \) and \( BA \) are both defined, then \( AB \) and \( BA \) are square matrices.
**b.** Prove: If \( A \) is an \( m \times n \) matrix and \( B \) is an \( n \times m \) matrix and \( A(BA) \) is defined, then \( B \) is an \( n \times m \) matrix.
### True-False Exercises
**TF.** In parts (a) - (o) determine whether the statement is true or false.
---
#### Explanation of Notation and Concepts:
- **Matrices:** Rectangular arrays of numbers arranged in rows and columns.
- **Square Matrix:** A matrix with the same number of rows and columns, i.e., an \( n \times n \) matrix.
- **Trace (tr):** The sum of the elements on the main diagonal of a square matrix.
##### Detailed Explanation of Problems:
- **Problem 35:** This involves proving that the trace of the sum of two \( n \times n \) matrices is equal to the sum of their traces. This is a fundamental property of the trace function in linear algebra.
- **Problem 36a:** This problem requires proving that if the products \( AB \) and \( BA \) are defined (meaning matrix multiplication is possible), both resulting matrices will be square matrices.
- **Problem 36b:** This involves proving a necessary condition for the dimensions of matrix \( B \) when matrix \( A \) is \( m \times n \), \( B \) is an \( n \times m \) matrix, and the product \( A(BA) \) is defined. This explores the implications of matrix multiplication on the dimensions of the matrices involved.
These exercises aim to deepen the understanding of the properties of matrix operations and the implications of these operations on the dimensions and structure of matrices.
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